Scalar dissipation and mean reaction rates in premixed turbulent combustion Ken Bray a , Michel Champion b,⇑ , Paul A. Libby c , N. Swaminathan a a Cambridge University, Cambridge CB2 1PZ, United Kingdom b Institut Pprime, Laboratoire de Combustion, UPR3346 CNRS ENSMA, 86961 Futuroscope, France c University of California San Diego La Jolla, CA 92093, United States article info Article history: Received 5 October 2010 Received in revised form 18 March 2011 Accepted 21 March 2011 Available online 18 April 2011 Keywords: Turbulent combustion modelling Scalar dissipation Flamelet regime abstract A general equation for a variance parameter, appearing as a crucial quantity in a simple algebraic expres- sion for the mean chemical rate, is derived. This derivation is based on a flamelet approach to model a turbulent premixed flame, for high but finite values of the Damköhler number. Application of this equa- tion to the case of a planar turbulent flame normal to the oncoming flow of reactants gives good agree- ment with DNS data corresponding to three different values of the Damköhler number and two values of the heat release parameter. Ó 2011 The Combustion Institute. Published by Elsevier Inc. All rights reserved. 1. Introduction Premixed turbulent combustion can be described in terms of a combustion progress variable c(x, t), with c = 0 in reactants and c = 1 in products, satisfying the equation @ @t ðqcÞþ r ðqucÞ¼ r ðqDrcÞþ x c ð1Þ where q, u and D represent gas density, flow velocity and diffusion coefficient, respectively, and x c is the chemical reaction rate [1]. Prediction of its mean value  x c either in RANS or LES calculations is one of the most difficult problems in turbulent combustion mod- elling [2,3]. One way to address this problem is through a presumed pdf model in which the mass-weighted pdf e P ðc; x; tÞ is assumed to have a specified shape controlled by the first and second Favre moments of c, so that e P ðc; x; tÞ¼ e P ðc;~ cðx; tÞ; f c 002 ðx; tÞÞ. An analysis of three different presumed pdf models [4] at large values of the Damköhler number Da = t t /t c , where t t and t c are char- acteristic turbulence and chemical time scales, leads to the simple result that  x c ¼  cc B  q~ cð1  ~ cÞI ð2Þ where B, in s 1 , is the constant coefficient of the pre-exponential factor in the global reaction rate expression, and I is an integral quantity defined later on (see Eq. (38)), whose value depends on the reaction rate expression as well as the shape of the selected pdf [4]. Although I is known for a given pdf, its value is uncertain to the extent that the shape of the pdf is an approximation. Also  cc is related to the variance f c 002 by  cc ¼ 1  f c 002 ~ cð1  ~ cÞ ð3Þ so that  cc is equal to unity if f c 002 ¼ 0 and is zero if f c 002 reaches its maximum possible value of ~ cð1  ~ cÞ. In the latter case, for which Da  1; e P ðc; x; tÞ is bimodal and consists only of delta functions at c = 0 and c = 1. Before Eq. (2 ) can be used it is necessary to derive either an expression for  cc or a balance equation for this quantity. The Favre mean ~ cðx; tÞ and variance f c 002 ðx; tÞ, which determine the shape of the presumed pdf e P ðc; x; tÞ, must be calculated from closed transport equations. To do so, in addition to several other closure problems, a model must be provided for the mean scalar dissipation which appears as a sink term in the variance equation. The mean scalar dissipation is a measure of the rate at which molecular diffusion processes lead to small-scale mixing in turbu- lent flows. It plays an important role in many theoretical descrip- tions [5–9] of the mean rate of chemical reaction in turbulent combustion, particularly when Da  1. If the fuel and oxidiser are supplied to the combustion zone separately, the scalar dissipa- tion describes the rate at which they mix and burn. In the case of premixed combustion the scalar dissipation represents the rate at which cold unburned reactants and hot fully burned products are mixed and burned. It is defined as e v ¼ qD$c 00  $c 00 =  q ð4Þ If Da  1, combustion is confined to thin propagating reaction zones, whose internal structure resembles that of a laminar flame, and which separate unburned reactants from combustion products. The composition gradient appearing in Eq. (4) is then related to that in a laminar flame. Theory [10,11] and DNS [12] both show that if Da  1 the scalar dissipation e v is proportional to the chemical 0010-2180/$ - see front matter Ó 2011 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2011.03.009 ⇑ Corresponding author. Fax: +33 549498176. E-mail address: michel.champion@lcd.ensma.fr (M. Champion). Combustion and Flame 158 (2011) 2017–2022 Contents lists available at ScienceDirect Combustion and Flame journal homepage: www.elsevier.com/locate/combustflame